Integrand size = 20, antiderivative size = 100 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
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Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1599, 1128, 715, 648, 632, 212, 642} \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 715
Rule 1128
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^7}{a+b x^2+c x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^3}{a+b x+c x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {a b+\left (b^2-a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {\text {Subst}\left (\int \frac {a b+\left (b^2-a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^2} \\ & = -\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}-\frac {\left (b \left (b^2-3 a c\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac {\left (b^2-a c\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3} \\ & = -\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac {\left (b \left (b^2-3 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3} \\ & = -\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\frac {c x^2 \left (-2 b+c x^2\right )-\frac {2 b \left (b^2-3 a c\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
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Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {-\frac {1}{2} c \,x^{4}+b \,x^{2}}{2 c^{2}}+\frac {\frac {\left (-a c +b^{2}\right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a b -\frac {\left (-a c +b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c^{2}}\) | \(105\) |
risch | \(\frac {x^{4}}{4 c}-\frac {b \,x^{2}}{2 c^{2}}+\frac {b^{2}}{4 c^{3}}-\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) a \,b^{2}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) b^{4}}{4 c^{3} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{4 c^{3} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) a \,b^{2}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) b^{4}}{4 c^{3} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{4 c^{3} \left (4 a c -b^{2}\right )}\) | \(957\) |
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Time = 0.27 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.13 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} - {\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} + 2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (92) = 184\).
Time = 1.75 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.91 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=- \frac {b x^{2}}{2 c^{2}} + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c - a b^{2} + 8 a c^{3} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c - a b^{2} + 8 a c^{3} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac {x^{4}}{4 c} \]
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\[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\int { \frac {x^{8}}{c x^{5} + b x^{3} + a x} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\frac {c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} + \frac {{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{3}} \]
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Time = 8.58 (sec) , antiderivative size = 842, normalized size of antiderivative = 8.42 \[ \int \frac {x^8}{a x+b x^3+c x^5} \, dx=\frac {x^4}{4\,c}-\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {b\,x^2}{2\,c^2}+\frac {b\,\mathrm {atan}\left (\frac {2\,c^4\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {b\,\left (3\,a\,c-b^2\right )\,\left (\frac {8\,a^2\,c^4-8\,a\,b^2\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )}{8\,c^3\,\sqrt {4\,a\,c-b^2}}-\frac {a\,b\,\left (3\,a\,c-b^2\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{c\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}}{a}-x^2\,\left (\frac {\frac {b\,\left (\frac {6\,b^3\,c^3-10\,a\,b\,c^4}{c^4}+\frac {4\,b\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (3\,a\,c-b^2\right )}{8\,c^3\,\sqrt {4\,a\,c-b^2}}+\frac {b^2\,\left (3\,a\,c-b^2\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,c\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}}{a}+\frac {b\,\left (\frac {2\,a^2\,b\,c^2-3\,a\,b^3\,c+b^5}{c^4}+\frac {\left (\frac {6\,b^3\,c^3-10\,a\,b\,c^4}{c^4}+\frac {4\,b\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {b^3\,{\left (3\,a\,c-b^2\right )}^2}{2\,c^4\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {\left (\frac {8\,a^2\,c^4-8\,a\,b^2\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {a^3\,c^2-2\,a^2\,b^2\,c+a\,b^4}{c^4}+\frac {a\,b^2\,{\left (3\,a\,c-b^2\right )}^2}{c^4\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{9\,a^2\,b^2\,c^2-6\,a\,b^4\,c+b^6}\right )\,\left (3\,a\,c-b^2\right )}{2\,c^3\,\sqrt {4\,a\,c-b^2}} \]
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